1. 1
   The graph of $y = f(x)$, where $f(x) = x^2 - 4x + k$ and $k$ is a constant, passes through the point $(1, 2)$.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$-3$
   • B$1$
   • C$5$
   • D$2$

   (b)

   Which of these points also lies on the graph of $y = f(x)$?

   [1 mark]

   • A$(-1, 2)$
   • B$(2, 2)$
   • C$(4, 1)$
   • D$(3, 2)$

   (c)

   Find the coordinates of the points where the graph of $y = f(x)$ meets the line $y = 10$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two water tanks are being emptied at the same time. The volume, in litres, of water in tank 1 after $t$ minutes is $V(t) = 800 - 50t$, for $0 \le t \le 16$. The volume in tank 2 is $W(t) = 500 - 20t$, for $0 \le t \le 25$.

   (a)

   Which statement correctly describes the graph of $y = V(t)$?

   [1 mark]

   • AA straight line segment rising from $(0, 0)$ to $(16, 800)$
   • BA straight line segment from $(0, 800)$ to $(16, 0)$
   • CA straight line segment from $(0, 16)$ to $(800, 0)$
   • DA straight line through the origin with gradient $-50$

   (b)

   Find the time at which tank 1 is one-quarter full.

   [1 mark]

   • A$12$ minutes
   • B$8$ minutes
   • C$4$ minutes
   • D$16$ minutes

   (c)

   Find the time at which the two tanks hold the same volume of water, and write down this volume.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A workshop makes and sells $x$ lamps per day, where $0 \le x \le 100$. Its daily revenue is $R(x) = 60x - 0.5x^2$ dollars and its daily cost is $C(x) = 400 + 12x$ dollars. The daily profit is $P(x) = R(x) - C(x)$.

   (a)

   Find an expression for $P(x)$ in the form $ax^2 + bx + c$, and interpret the value of $P(0)$ in context.

   [3 marks]

   (b)

   Use technology to find the numbers of lamps the workshop can make and sell in a day in order to make a profit, and find the maximum daily profit.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two seedlings are measured every week. The height, in cm, of seedling A after $t$ weeks is $a(t) = 2 + 3t$ and the height of seedling B is $b(t) = 0.5t^2 + 1$, for $0 \le t \le 8$. The function $d$ gives how much taller seedling A is than seedling B: $d(t) = a(t) - b(t)$.

   (a)

   (i) Show that $d(t) = 1 + 3t - 0.5t^2$.
   (ii) Find the value of $t$ at which seedling A is furthest ahead of seedling B, and find how much taller it is at that time.
   (iii) Use technology to find the value of $t$ at which the two seedlings have the same height.

   [6 marks]

   (b)

   Without drawing a graph, describe the graph of $y = d(t)$ for $0 \le t \le 8$ by writing down the value of $d(0)$ and the value of $d(8)$ and stating the range of $d$. Interpret the sign of $d(t)$ in context for $0 \le t < 6.32$ and for $6.32 < t \le 8$.

   [6 marks]

   Total for question 4: 12 marks

End of questions